Tagged Questions

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How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
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quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
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Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
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Using a theorem to find the center of a $p$-sylow subgroup of simple group

I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown ...
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Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
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regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
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Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$ Hint: consider the action of $G$ on right cosets of $H$ in $G$. I'm ...
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What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
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A question about the involution in simple groups.

Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question ...
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Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
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Size of maximal tori in finite simple groups of Lie type

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$? ...
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Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
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Existance of semisimple elements in a torus.

Let $G$ be a finite simple group of Lie type and $T$ be a maximal torus in $G$. Is it true that $T$ contains a regular semisimple element (a semisimple element which $C_G(x)=T$)? If yes, why?
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If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
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dimension of a finie simple group of Lie type.

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$, namely $F_q$ of size $q$. Suppose $dim(G)=n$. What we can say about $|G|$? Is it true that $|G|=q^ns$, where ...
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Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
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Brauer characters of finite simpl groups of type E8

I would like to know the Brauer characters of finite simple groups $E_8(2)$, $E_8(3)$ or $E_8(5)$. Is there any refrence for this topic? Thanks
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Automorphisms of spin groups over finite fields, even dimension

I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of ...
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Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
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Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy ...
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I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
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Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
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Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
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Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...
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Prove that $G=S_{1}\times S_{2}\times\cdots\times S_{k}$

Let $G$ be a finite group and let $N_1,N_2,\ldots,N_k$ be normal subgroups. Suppose that $\bigcap_{i=1}^{k}N_{i}=\{1\}$ and that $G/N_i=S_i$ is simple. If all the groups $S_i$ are non-isomorphic ...
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Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
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Groups of order $56$

Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove? My idea: ...
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No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
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A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
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Simple group with order $\geq n!$ cannot have subgroup of index $n$.

My problem is as seen in the title: For positive integer $n>1$, prove that a simple group with order $\geq n!$ cannot have subgroup of index $n$. Could anyone give me some hints on how to ...
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Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
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non-split extension of the simple group $L_3(4)$

I would like to know the structure of the groups $L_3(4).C_2$ and $L_3(4).C_{11}$. (By $C_n$ I mean the cyclic group of order $n$ and by $G=K.L$ I mean the non-spli extension of $K$ by $L$, were $K$ ...
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Are quasinilpotent groups a Fitting class?

A finite group is called quasinilpotent if it induces inner automorphisms on all of its chief factors. A solvable group is quasinilpotent iff it centralizes all of its (necessarily abelian) chief ...
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Are there any distinct finite simple groups with the same order?

In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two ...
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The Monster group

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
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Projective linear group - solvable

Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
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Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
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How confident can we be about the validity of the classification of finite simple groups?

The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed. How certain can we be that The proof of ...